Properties

Label 1344.4.a.bj
Level $1344$
Weight $4$
Character orbit 1344.a
Self dual yes
Analytic conductor $79.299$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,4,Mod(1,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(79.2985670477\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 672)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{17}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + ( - \beta + 5) q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + ( - \beta + 5) q^{5} + 7 q^{7} + 9 q^{9} + (11 \beta + 11) q^{11} + ( - 2 \beta + 12) q^{13} + (3 \beta - 15) q^{15} + (11 \beta - 75) q^{17} + ( - 32 \beta - 4) q^{19} - 21 q^{21} + ( - 33 \beta - 41) q^{23} + ( - 10 \beta - 83) q^{25} - 27 q^{27} + (32 \beta + 18) q^{29} + ( - 20 \beta + 44) q^{31} + ( - 33 \beta - 33) q^{33} + ( - 7 \beta + 35) q^{35} + (14 \beta - 144) q^{37} + (6 \beta - 36) q^{39} + (61 \beta - 193) q^{41} + (76 \beta + 172) q^{43} + ( - 9 \beta + 45) q^{45} + (110 \beta - 138) q^{47} + 49 q^{49} + ( - 33 \beta + 225) q^{51} + ( - 18 \beta - 80) q^{53} + (44 \beta - 132) q^{55} + (96 \beta + 12) q^{57} + (46 \beta + 538) q^{59} + ( - 148 \beta - 78) q^{61} + 63 q^{63} + ( - 22 \beta + 94) q^{65} + (6 \beta + 686) q^{67} + (99 \beta + 123) q^{69} + ( - 55 \beta - 551) q^{71} + ( - 178 \beta - 120) q^{73} + (30 \beta + 249) q^{75} + (77 \beta + 77) q^{77} + (106 \beta - 206) q^{79} + 81 q^{81} + ( - 164 \beta + 232) q^{83} + (130 \beta - 562) q^{85} + ( - 96 \beta - 54) q^{87} + ( - 51 \beta - 873) q^{89} + ( - 14 \beta + 84) q^{91} + (60 \beta - 132) q^{93} + ( - 156 \beta + 524) q^{95} + (50 \beta + 428) q^{97} + (99 \beta + 99) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 22 q^{11} + 24 q^{13} - 30 q^{15} - 150 q^{17} - 8 q^{19} - 42 q^{21} - 82 q^{23} - 166 q^{25} - 54 q^{27} + 36 q^{29} + 88 q^{31} - 66 q^{33} + 70 q^{35} - 288 q^{37} - 72 q^{39} - 386 q^{41} + 344 q^{43} + 90 q^{45} - 276 q^{47} + 98 q^{49} + 450 q^{51} - 160 q^{53} - 264 q^{55} + 24 q^{57} + 1076 q^{59} - 156 q^{61} + 126 q^{63} + 188 q^{65} + 1372 q^{67} + 246 q^{69} - 1102 q^{71} - 240 q^{73} + 498 q^{75} + 154 q^{77} - 412 q^{79} + 162 q^{81} + 464 q^{83} - 1124 q^{85} - 108 q^{87} - 1746 q^{89} + 168 q^{91} - 264 q^{93} + 1048 q^{95} + 856 q^{97} + 198 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.56155
−1.56155
0 −3.00000 0 0.876894 0 7.00000 0 9.00000 0
1.2 0 −3.00000 0 9.12311 0 7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.a.bj 2
4.b odd 2 1 1344.4.a.br 2
8.b even 2 1 672.4.a.j yes 2
8.d odd 2 1 672.4.a.e 2
24.f even 2 1 2016.4.a.q 2
24.h odd 2 1 2016.4.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.a.e 2 8.d odd 2 1
672.4.a.j yes 2 8.b even 2 1
1344.4.a.bj 2 1.a even 1 1 trivial
1344.4.a.br 2 4.b odd 2 1
2016.4.a.q 2 24.f even 2 1
2016.4.a.r 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5}^{2} - 10T_{5} + 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 22T_{11} - 1936 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 22T - 1936 \) Copy content Toggle raw display
$13$ \( T^{2} - 24T + 76 \) Copy content Toggle raw display
$17$ \( T^{2} + 150T + 3568 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T - 17392 \) Copy content Toggle raw display
$23$ \( T^{2} + 82T - 16832 \) Copy content Toggle raw display
$29$ \( T^{2} - 36T - 17084 \) Copy content Toggle raw display
$31$ \( T^{2} - 88T - 4864 \) Copy content Toggle raw display
$37$ \( T^{2} + 288T + 17404 \) Copy content Toggle raw display
$41$ \( T^{2} + 386T - 26008 \) Copy content Toggle raw display
$43$ \( T^{2} - 344T - 68608 \) Copy content Toggle raw display
$47$ \( T^{2} + 276T - 186656 \) Copy content Toggle raw display
$53$ \( T^{2} + 160T + 892 \) Copy content Toggle raw display
$59$ \( T^{2} - 1076 T + 253472 \) Copy content Toggle raw display
$61$ \( T^{2} + 156T - 366284 \) Copy content Toggle raw display
$67$ \( T^{2} - 1372 T + 469984 \) Copy content Toggle raw display
$71$ \( T^{2} + 1102 T + 252176 \) Copy content Toggle raw display
$73$ \( T^{2} + 240T - 524228 \) Copy content Toggle raw display
$79$ \( T^{2} + 412T - 148576 \) Copy content Toggle raw display
$83$ \( T^{2} - 464T - 403408 \) Copy content Toggle raw display
$89$ \( T^{2} + 1746 T + 717912 \) Copy content Toggle raw display
$97$ \( T^{2} - 856T + 140684 \) Copy content Toggle raw display
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